Since its introduction the consistency around the cube (CAC) has been a source of many results in the classification of nonlinear integrable partial difference equations on a quad graph. The importance of this criterion relies on the fact that it ensures the existence of Bäcklund transformations [1]}, [2]}, [3]}, [4]}, [5]} and, as a consequence, of Lax pairs. However [6]} Lax pairs and Bäcklund transforms are associated with both linearizable and integrable equations. Let us point out that to be bona fide a Lax pair has to give rise to a genuine spectral problem [7]}, otherwise the Lax pair is a fake Lax pair [8]}, [9]}, [10]}, [11]}, [12]}. A fake Lax pair is useless in proving (or disproving) the integrability, since it can be equally found for integrable and non-integrable equations. In the linearizable case Lax pairs are fake, even though proving this it is usually nontrivial [13]}.
